In [1]} the mapping \(h^* = \big [E_1^{\mathrm {\mathbf {Bis}}}(h)\big ]^- \colon G \rightarrow F\)  is the ring homomorphism
induced by \(h \in \mathrm {Mor}(F̏, G̏)\) .
Let the Chu space \((X, \mathrm {ev}_{{X,F}}, F)\) , and by Definition REF  let \(\widehat{\mathrm {ev}_{{X,F}}} \colon X \rightarrow (F \rightarrow {\mathbb {R}})\) 
with \(\widehat{\mathrm {ev}_{{X,F}}}(x) = \widehat{x}\) .
Consequently, the Chu space \((X, \mathrm {ev}_{{X,F}}, F)\) 
is separable if and only if \(F\)  separates the points of \(X\) :
\(\widehat{x} =_{F̥(F, {\mathbb {R}})} \widehat{x{^{\prime }}} & :\Leftrightarrow \forall _{f \in F}\big (\widehat{x}(f) =_{{\mathbb {R}}} \widehat{x{^{\prime }}}(f)\big )\\& \Leftrightarrow \forall _{f \in F}\big (f(x) =_{{\mathbb {R}}} f(x{^{\prime }})\big )\\& \Leftrightarrow x =_X x{^{\prime }}.\) 
